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1% $Id: basic_equations.tex 1531 2015-01-26 13:58:29Z keck $
2\input{header_tmp.tex}
3%\input{../header_LECTURE.tex}
4
5\usepackage[utf8]{inputenc}
6\usepackage{ngerman}
7\usepackage{pgf}
8\usepackage{subfigure}
9\usepackage{units}
10\usepackage{multimedia}
11\usepackage{hyperref}
12\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
13\usepackage{xmpmulti}
14\usepackage{tikz}
15\usetikzlibrary{shapes,arrows,positioning,decorations.pathreplacing}
16\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
17
18%---------- neue Pakete
19\usepackage{amsmath}
20\usepackage{amssymb}
21\usepackage{multicol}
22\usepackage{pdfcomment}
23\usepackage{xcolor}
24
25\institute{Institute of Meteorology and Climatology, Leibniz UniversitÀt Hannover}
26\selectlanguage{english}
27\date{last update: \today}
28\event{PALM Seminar}
29\setbeamertemplate{navigation symbols}{}
30\setbeamersize{text margin left=.5cm,text margin right=.2cm}
31\setbeamertemplate{footline}
32  {%
33    \begin{beamercolorbox}[rightskip=-0.1cm]&
34     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
35    \end{beamercolorbox}
36    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,%
37      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}%
38      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}%
39    \end{beamercolorbox}%
40%    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}%
41%    \end{beamercolorbox}
42  }%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.eps}}
43
44\title[Basic Equations]{Basic Equations}
45\author{PALM group}
46
47\begin{document}
48
49%Folie 1
50\begin{frame}
51   \titlepage
52\end{frame}
53
54
55\section{Basic equations}
56\subsection{Basic equations, Unfiltered}
57
58% Folie 2
59\begin{frame}
60   \frametitle{Basic equations, Unfiltered}
61   \setlength{\leftmargini}{0.3cm}
62   \begin{itemize}
63       \item<2->Navier-Stokes equations
64      \begin{equation*}
65         \rho \frac{\partial u_i}{\partial t} + \rho u_k
66         \frac{\partial u_i}{\partial x_k} =
67         - \frac{\partial p}{\partial x_i} - \rho \varepsilon_{ijk} 
68         f_j u_k - \rho \frac{\partial \phi}{\partial x_i} + \mu 
69         \left\{ \frac{\partial^2 u_i}{\partial x_k^2} + \frac{1}{3} 
70         \frac{\partial}{\partial x_i} \left(
71         \frac{\partial u_k}{\partial x_k} \right) \right\}
72      \end{equation*}
73      \item \onslide<3->First principle
74      \begin{equation*}
75         \rho \frac{\partial T}{\partial t} + \rho u_k \frac{\partial T}{\partial x_k} = \mu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q
76      \end{equation*}
77      \item \onslide<4->Equation for passive scalar
78      \begin{equation*}
79         \rho \frac{\partial \psi}{\partial t} + \rho u_k \frac{\partial \psi}{\partial x_k} = \mu_{\psi} \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi}
80      \end{equation*}
81      \item \onslide<5->Continuity equation
82      \begin{equation*}
83         \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} 
84      \end{equation*}
85   \end{itemize}
86\end{frame}
87
88% Folie 3
89\begin{frame}
90   \frametitle{Boussinesq Approximation}
91   \footnotesize
92   \begin{itemize}
93      \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$ 
94      \begin{align*}
95         T(x,y,z,t) &= T_0(x,y,z) &+& T^{*}(x,y,z,t)&&\\
96         p(x,y,z,t) &= p_0(x,y,z) &+& p^{*}(x,y,z,t)&&\\
97         \rho(x,y,z,t) &= \rho_0(z) &+& \rho^{*}(x,y,z,t);& &
98         &\psi^{*} << \psi_0&
99      \end{align*} 
100      \item \onslide<3->Hydrostatic equilibrium, geostrophic wind (not included in Boussinesq)
101      \begin{equation*}
102         \frac{\partial p_0}{\partial z} = -g \rho_0 \hspace{10mm} 
103         \frac{1}{\rho_0} \frac{\partial p_0}{\partial x} = -f v_\mathrm{g},
104         \hspace{5mm} \frac{1}{\rho_0} \frac{\partial p_0}{\partial y} = f u_\mathrm{g}
105      \end{equation*}
106      \item \onslide<4->Equation of state
107      \begin{equation*}
108         p = \rho R T \rightarrow \ln{p} = \ln{\rho} + \ln{R} + \ln{T} \rightarrow \frac{d p}{p} = \frac{d \rho}{\rho} + \frac{d T}{T} 
109      \end{equation*}
110      \begin{equation*}
111         \frac{\Delta p}{p_0} \approx \frac{\Delta \rho}{\rho_0} +
112         \frac{\Delta T}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx 
113         \frac{\rho^{*}}{\rho_0} + \frac{T^{*}}{T_0} \hspace{10mm} 
114         \frac{\rho^{*}}{\rho_0} \approx - \frac{T^{*}}{T_0} \hspace{10mm}
115      \end{equation*}
116   \end{itemize}
117\end{frame}
118
119% Folie 4
120\begin{frame}
121   \frametitle{Continuity Equation}
122   \begin{eqnarray*}
123      \onslide<2-> \dfrac{\partial \rho_0(z)}{\partial t} =
124      - \dfrac{\partial \rho_0(z) u_k}{\partial x_k} & 
125      \hspace{10mm} \dfrac{\partial \rho_0 u_k}{\partial x_k} = 0
126      \hspace{5mm} & \text{anelastic approximation}\\
127      \\
128      \onslide<3-> \rho_0 = const. & \hspace{10mm} 
129      \dfrac{\partial u_k}{\partial x_k} = 0 \hspace{5mm} & 
130      \text{incompressible flow}
131   \end{eqnarray*}
132\end{frame}
133
134% Folie 5
135\begin{frame}
136   \frametitle{Boussinesq Approximated Equations}
137   \begin{itemize}
138      \item \onslide<2->Navier-Stokes equations
139      \begin{equation*}
140         \frac{\partial u_i}{\partial t} 
141         + \frac{\partial u_k u_i}{\partial x_k} = 
142         - \frac{1}{\rho_0}\frac{\partial p^{*}}{\partial x_i} 
143         - \varepsilon_{ijk} f_j u_k + \varepsilon_{i3k} f_3 u_{k_\mathrm{g}} 
144         + g \frac{T - T_0}{T_0} \delta_{i3} + \nu 
145         \frac{\partial^2 u_i}{\partial x_k^2}
146      \end{equation*}
147      \item \onslide<3->First principle
148      \begin{equation*}
149         \frac{\partial T}{\partial t} + u_k \frac{\partial T}{\partial x_k} =
150         \nu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q
151      \end{equation*}
152      \item \onslide<4->Equation for passive scalar
153      \begin{equation*}
154         \frac{\partial \psi}{\partial t} + u_k
155         \frac{\partial \psi}{\partial x_k} = \nu_{\psi} 
156         \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi}
157      \end{equation*}
158      \item \onslide<5->Continuity equation
159      \begin{equation*}
160         \frac{\partial u_k}{\partial x_k} = 0
161      \end{equation*}
162   \end{itemize}
163   \onslide<6-> \tikzstyle{plain} = [rectangle, draw, text width=0.255\textwidth, font=\small]
164
165   \begin{tikzpicture}[remember picture, overlay]
166      \node at (current page.north west){%
167      \begin{tikzpicture}[overlay]
168         \node[plain, draw,anchor=west] at (94mm,-55mm) {\noindent This set of equations is valid for almost all kind of CFD models!};
169      \end{tikzpicture}
170      };
171   \end{tikzpicture}
172\end{frame}
173
174
175\section{Scale Separation}
176\subsection{Scale Separation by Spatial Filtering}
177
178% Folie 6
179\begin{frame}
180   \frametitle{LES - Scale Separation by Spatial Filtering (I)}
181   \footnotesize
182   \begin{itemize}
183      \item<1->{LES technique is based on scale separation, in order to reduce the number of degrees of freedom of the solution. \begin{math} \boxed{\Psi(x_i , t) = \overline{\Psi}(x_i , t) + \Psi'(x_i , t)} \end{math}}
184      \item<2->{Large / low-frequency modes $\Psi$ are calculated directly (resolved scales).}
185      \item<3->{Small / high-frequency modes $\Psi'$ are parameterized using a statistical model (subgrid / subfilter scales, SGS model).}
186      \item<4->{These two categories of scales are seperated by defining a cutoff length $\Delta$.}
187   \end{itemize}
188   \normalsize
189   \includegraphics[width=\textwidth]{basic_equations_figures/Spatial_Filtering_I.png}
190\end{frame}
191
192
193
194% Folie 7
195\begin{frame}
196   \frametitle{LES - Scale Separation by Spatial Filtering (II)}
197   \begin{columns}[T]
198      \begin{column}{0.8\textwidth}
199      \footnotesize
200      \begin{itemize}
201         \item<1->The Filter applied is a spatial filter:
202         \begin{equation*} 
203            \overline{\Psi}(x_i) = \int_D G(x_i - x_i') \Psi(x_i')dx_i'
204         \end{equation*}
205         \begin{equation*} 
206            \overline{\Psi}'(x_i) = 0 \qquad but \qquad \overline{\overline{\Psi}} \neq \overline{\Psi}(x_i)
207         \end{equation*} 
208         \item<2->Filter applied to the nonlinear advection term:
209         \begin{equation*} 
210            \overline{u_k u_i} = \overline{(\overline{u_k} + u_k')(\overline{u_i} + u_i')} = \overline{\overline{u_k}\,\overline{u_i}} + \underbrace{\overline{\overline{u_k}          u_i'} + \overline{u_k' \overline{u_i}}}_{C_{ki}} + \underbrace{\overline{u_k' u_i'}}_{R_{ki}} 
211         \end{equation*}   
212         \item<5->Leonard proposes a further decomposition:
213         \begin{equation*} 
214            \overline{\overline{u_k}\,\overline{u_i}} = \overline{u_k}\,\overline{u_i} + \underbrace{\left( \overline{\overline{u_k}\,\overline{u_i}} - \overline{u_k}\,\overline{u_i} \right)}_{L_{ki}}                         
215         \end{equation*}
216         \begin{equation*} 
217            \overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + L_{ki} + C_{ki} + R_{ki} = \overline{u_k}\,\overline{u_i} + \tau_{ki} 
218         \end{equation*}       
219      \end{itemize}
220      \end{column}
221      \begin{column}{0.32\textwidth}
222      \vspace{45mm}
223      \hspace{-1.75cm}
224      \begin{footnotesize}
225         \onslide<3->$R_{ki}$: \textbf{Reynolds-stress} \\
226         \hspace*{-1.5cm}$C_{ki}$: \textbf{cross-stress} \\
227         \hspace*{-1.5cm}$L_{ki}$: \textbf{Leonard-stress} \\
228         \hspace*{-1.5cm}$\tau_{ki}$: \textbf{total stress-tensor}\\
229         \hspace*{-1.05cm} \textbf{generalized Reynolds stress}\\
230      \end{footnotesize}
231      \end{column}
232   \end{columns}
233   \onslide<4->\tikzstyle{plain} = [rectangle, draw, text width=0.25\textwidth, font=\small]
234      \begin{tikzpicture}[remember picture, overlay]
235      \node at (current page.north west){
236      \begin{tikzpicture}[overlay]
237         \node[plain, draw,anchor=west] at (94mm,-30mm) {
238         \begin{footnotesize}
239            \noindent \textbf{Ensemble average:} \\
240         \end{footnotesize}
241         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
242         \vspace{5mm}
243         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
244         };
245      \end{tikzpicture}
246      };
247   \end{tikzpicture}   
248\end{frame}
249
250% Folie 8
251\begin{frame}
252   \frametitle{LES - Scale Separation by Spatial Filtering (III)}
253   \small
254   \begin{itemize}
255      \item<2-> Volume-balance approach (Schumann, 1975)\\ advantage: numerical discretization acts as a\\ Reynolds operator
256      \begin{flalign*}
257         &\Psi(V,t)=\frac{1}{\Delta x \cdot \Delta y \cdot \Delta z} = \int \int \int_V \Psi(V',t) dV'&\\
258         &\overline{\Psi'}(x_i)=0 \hspace{5mm} \text{and} \hspace{5mm} \overline{\overline{\Psi}} = \overline{\Psi}\\
259         &V=\left[ x - \frac{\Delta x}{2}, x + \frac{\Delta x}{2} \right] \times \left[ y - \frac{\Delta y}{2}, y + \frac{\Delta y}{2} \right] \times \left[ z - \frac{\Delta z}{2}, z + \frac{\Delta z}{2} \right]
260      \end{flalign*}
261      \item<3-> Filter applied to the nonlinear advection term:
262      \begin{equation*}
263         \overline{u_k u_i} = \overline{(\overline{u_k}+u'_k)(\overline{u_i}+u'_i)}=
264         \overline{u_k}\,\overline{u_i}+\overline{u'_k u'_i}
265      \end{equation*}
266   \end{itemize}
267   \onslide<1->\tikzstyle{plain} = [rectangle, draw, text width=0.25\textwidth, font=\small]
268      \begin{tikzpicture}[remember picture, overlay]
269      \node at (current page.north west){
270      \begin{tikzpicture}[overlay]
271         \node[plain, draw,anchor=west] at (94mm,-30mm) {
272         \begin{footnotesize}
273            \noindent \textbf{Ensemble average:} \\
274         \end{footnotesize}
275         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
276         \vspace{5mm}
277         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
278         };
279      \end{tikzpicture}
280      };
281   \end{tikzpicture}
282 \end{frame}
283
284
285\section{Filtered equations}
286\subsection{The Filtered Equations}
287
288% Folie 9
289\begin{frame}
290   \frametitle{The Filtered Equations}
291   \onslide<2->
292   \begin{equation*}
293      \frac{\partial \overline{u_i}}{\partial t} 
294      + \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} =
295      - \frac{1}{\rho_0} \frac{\partial \overline{p}^*}{\partial x_i} 
296      - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} 
297      + g \frac{\overline{T}-T_0}{T_0} \delta_{i3} 
298      + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} 
299      - \frac{\partial \tau_{ki}}{\partial x_k}
300   \end{equation*}
301
302   \begin{footnotesize}
303      \begin{itemize}
304         \item<3->The previous derivation completely ignores the existance of the computational grid.
305         \item<4->The computational grid introduces another space scale: the discretization step $\Delta x_i$.
306         \item<5->$\Delta x_i$ has to be small enough to be able to apply the filtering process correctly: $\Delta x_i \le \Delta$
307         \item<6-> Two possibilities:\\
308         1. Pre-filtering technique\\
309         ($\Delta x < \Delta$,  explicit filtering)\\
310         2. Linking the analytical filter\\
311         to the computational grid\\
312         ($\Delta x = \Delta$, implicit filtering)
313      \end{itemize}
314   \end{footnotesize}
315
316   \begin{picture}(0.0,0.0)
317      \put(140,0){\uncover<6->{\includegraphics[width=0.6\textwidth]{basic_equations_figures/explicit_implicit.png}}}
318   \end{picture}
319\end{frame}
320
321%% Folie 10
322\begin{frame}
323  \frametitle{Explicit Versus Implicit Filtering}
324   \begin{itemize}
325      \item<2-> Explicit filtering:
326      \begin{itemize}
327         \small
328         \item<2-> Requires that the analytical filter is applied explicitly.
329         \item<3-> Rarely used in practice, due to additional computational costs.
330      \end{itemize}
331      \item<4-> Implicit filtering:
332      \begin{itemize}
333         \small
334         \item<4-> The analytical cutoff length is associated with the grid spacing.
335         \item<5-> This method does not require the use of an analytical filter.
336         \item<6-> The filter characteristic cannot really be controlled.
337         \item<7-> Because of its simplicity, this method is used by nearly all LES models.
338      \end{itemize}
339   \end{itemize}
340
341   \onslide<8->
342   \begin{scriptsize}
343      \textbf{Literature:}\\
344      \textbf{Sagaut, P., 2001:} Large eddy simulation for incompressible flows: An introduction. Springer Verlag, Berlin/Heidelberg/New York, 319 pp.\\
345      \textbf{Schumann, U., 1975:} Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys., \textbf{18}, 376-404.\\
346   \end{scriptsize}
347\end{frame}
348
349% Folie 11
350\begin{frame}
351   \frametitle{The Final Set of Equations (PALM)}
352   \footnotesize
353   \begin{itemize}
354      \item<2-> Navier-Stokes equations:
355      \onslide<2->
356      \begin{flalign*}
357         &\frac{\partial \overline{u_i}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} - \frac{1}{\rho_0} \frac{\partial \overline{\pi}^*}{\partial x_i} - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} + g \frac{\overline{\theta}-\theta_0}{\theta_0} \delta_{i3} + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} - \frac{\partial \tau_{ki}^r}{\partial x_k}&
358      \end{flalign*}
359      \item<4-> First principle (using potential\\ temperature):
360      \onslide<4->
361      \begin{flalign*}
362         &\frac{\partial \overline{\theta}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{\theta}}{\partial x_k} - \frac{\partial H_k}{\partial x_k} + Q_{\theta}&
363      \end{flalign*}
364      \item<5-> Equation for specific humidity\\ (passive scalar)
365      \onslide<5->
366      \begin{flalign*}
367         &\frac{\partial \overline{q}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{q}}{\partial x_k} - \frac{\partial W_k}{\partial x_k} + Q_{w}&
368      \end{flalign*}
369      \item<6-> 
370      Continuity equation
371      \onslide<6->
372      \begin{flalign*}
373         &\frac{\partial \overline{u_k}}{\partial x_k} = 0&
374      \end{flalign*}
375   \end{itemize}
376
377   \onslide<3->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
378   \begin{tikzpicture}[remember picture, overlay]
379      \node at (current page.north west){
380      \begin{tikzpicture}[overlay]
381         \node[plain, draw,anchor=west] at (75mm,-45mm) {
382         \begin{tiny}
383            \noindent normal stresses included in the stress tensor are now included in a modified dynamic pressure:\\
384         \end{tiny}
385         $\tau_{ki}^r = \tau_{ki} - \frac{1}{3} \tau_{jj} \delta_{ki}$\\
386         \vspace{1mm}
387         $\overline{\pi}^* = \overline{p}^* + \frac{1}{3} \tau_{jj} \delta_{ki}$
388         };
389      \end{tikzpicture}
390      };
391   \end{tikzpicture}
392
393   \onslide<7->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
394   \begin{tikzpicture}[remember picture, overlay]
395      \node at (current page.north west){
396      \begin{tikzpicture}[overlay]
397         \node[plain, draw,anchor=west] at (75mm,-70mm) {
398         \begin{tiny}
399            \noindent subgrid-scale stresses (fluxes) to be parameterized in the SGS model:\\
400         \end{tiny}
401         $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\
402         $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\
403         $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$
404         };
405      \end{tikzpicture}
406      };
407   \end{tikzpicture}
408\end{frame}
409
410
411\end{document}
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